\(\int \frac {x^3 (d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [103]

Optimal result

Integrand size = 40, antiderivative size = 433 \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 a^3 e^3 \left (c d^2-a e^2\right ) (d+e x)}{c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (3 c^3 d^6+13 a c^2 d^4 e^2-187 a^2 c d^2 e^4+187 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^5 d^5 e^2}+\frac {\left (c^2 d^4-34 a c d^2 e^2+41 a^2 e^4\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 c^4 d^4 e}+\frac {\left (3 c d^2-5 a e^2\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {e x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2}+\frac {3 \left (c d^2-a e^2\right ) \left (c^3 d^6+5 a c^2 d^4 e^2+35 a^2 c d^2 e^4-105 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{11/2} d^{11/2} e^{5/2}} \] Output:

2*a^3*e^3*(-a*e^2+c*d^2)*(e*x+d)/c^5/d^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) 
^(1/2)-1/64*(187*a^3*e^6-187*a^2*c*d^2*e^4+13*a*c^2*d^4*e^2+3*c^3*d^6)*(a* 
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/e^2+1/32*(41*a^2*e^4-34*a*c*d 
^2*e^2+c^2*d^4)*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e+1/8*(- 
5*a*e^2+3*c*d^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3+1/4*e 
*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2+3/64*(-a*e^2+c*d^2)*( 
-105*a^3*e^6+35*a^2*c*d^2*e^4+5*a*c^2*d^4*e^2+c^3*d^6)*arctanh(c^(1/2)*d^( 
1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(11/2)/d^( 
11/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 10.87 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {(d+e x) \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (315 a^4 e^7-105 a^3 c d e^5 (3 d-e x)+a^2 c^2 d^2 e^3 \left (13 d^2-119 d e x-42 e^2 x^2\right )+c^4 d^4 x \left (3 d^3-2 d^2 e x-24 d e^2 x^2-16 e^3 x^3\right )+a c^3 d^3 e \left (3 d^3+11 d^2 e x+44 d e^2 x^2+24 e^3 x^3\right )\right )-3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (c^3 d^6+5 a c^2 d^4 e^2+35 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a e+c d x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{64 c^{11/2} d^{11/2} e^{5/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(x^3*(d + e*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2), 
x]
 

Output:

-1/64*((d + e*x)*(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a* 
e^2)]*(315*a^4*e^7 - 105*a^3*c*d*e^5*(3*d - e*x) + a^2*c^2*d^2*e^3*(13*d^2 
 - 119*d*e*x - 42*e^2*x^2) + c^4*d^4*x*(3*d^3 - 2*d^2*e*x - 24*d*e^2*x^2 - 
 16*e^3*x^3) + a*c^3*d^3*e*(3*d^3 + 11*d^2*e*x + 44*d*e^2*x^2 + 24*e^3*x^3 
)) - 3*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 35*a^2*c 
*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*e + c*d*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e] 
*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(c^(11/2)*d^(11/2)* 
e^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*Sqrt[(a*e + c*d*x)*(d + e*x) 
])
 

Rubi [A] (verified)

Time = 2.62 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1211, 2192, 27, 2192, 27, 2192, 27, 1160, 1092, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 (d+e x)^3}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1211

\(\displaystyle \frac {\int \frac {c^4 d^4 x^4 e^8+a^2 \left (c d^2-a e^2\right )^2 e^8+c^3 d^3 \left (2 c d^2-a e^2\right ) x^3 e^7-a c d \left (c d^2-a e^2\right )^2 x e^7+c^2 d^2 \left (c d^2-a e^2\right )^2 x^2 e^6}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {8 a^2 c d \left (c d^2-a e^2\right )^2 e^9+3 c^4 d^4 \left (3 c d^2-5 a e^2\right ) x^3 e^8-8 a c^2 d^2 \left (c d^2-a e^2\right )^2 x e^8+2 c^3 d^3 \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x^2 e^7}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {8 a^2 c d \left (c d^2-a e^2\right )^2 e^9+3 c^4 d^4 \left (3 c d^2-5 a e^2\right ) x^3 e^8-8 a c^2 d^2 \left (c d^2-a e^2\right )^2 x e^8+2 c^3 d^3 \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x^2 e^7}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (16 a^2 c^2 d^2 \left (c d^2-a e^2\right )^2 e^{10}-4 a c^3 d^3 \left (7 c^2 d^4-13 a c e^2 d^2+4 a^2 e^4\right ) x e^9+c^4 d^4 \left (c^2 d^4-34 a c e^2 d^2+41 a^2 e^4\right ) x^2 e^8\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {16 a^2 c^2 d^2 \left (c d^2-a e^2\right )^2 e^{10}-4 a c^3 d^3 \left (7 c^2 d^4-13 a c e^2 d^2+4 a^2 e^4\right ) x e^9+c^4 d^4 \left (c^2 d^4-34 a c e^2 d^2+41 a^2 e^4\right ) x^2 e^8}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {c^3 d^3 e^8 \left (2 a e \left (c^3 d^6-66 a c^2 e^2 d^4+105 a^2 c e^4 d^2-32 a^3 e^6\right )+c d \left (3 c^3 d^6+13 a c^2 e^2 d^4-187 a^2 c e^4 d^2+187 a^3 e^6\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^3 d^3 e^7 x \left (41 a^2 e^4-34 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c^3 d^3 e^7 x \left (41 a^2 e^4-34 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} c^2 d^2 e^7 \int \frac {2 a e \left (c^3 d^6-66 a c^2 e^2 d^4+105 a^2 c e^4 d^2-32 a^3 e^6\right )+c d \left (3 c^3 d^6+13 a c^2 e^2 d^4-187 a^2 c e^4 d^2+187 a^3 e^6\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c^3 d^3 e^7 x \left (41 a^2 e^4-34 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} c^2 d^2 e^7 \left (\frac {\left (187 a^3 e^6-187 a^2 c d^2 e^4+13 a c^2 d^4 e^2+3 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (c d^2-a e^2\right ) \left (-105 a^3 e^6+35 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}\right )}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c^3 d^3 e^7 x \left (41 a^2 e^4-34 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} c^2 d^2 e^7 \left (\frac {\left (187 a^3 e^6-187 a^2 c d^2 e^4+13 a c^2 d^4 e^2+3 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (c d^2-a e^2\right ) \left (-105 a^3 e^6+35 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}\right )}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}+\frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 a^3 e^3 (d+e x) \left (c d^2-a e^2\right )}{c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\frac {\frac {\frac {1}{2} c^3 d^3 e^7 x \left (41 a^2 e^4-34 a c d^2 e^2+c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-\frac {1}{4} c^2 d^2 e^7 \left (\frac {\left (187 a^3 e^6-187 a^2 c d^2 e^4+13 a c^2 d^4 e^2+3 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {3 \left (c d^2-a e^2\right ) \left (-105 a^3 e^6+35 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}\right )}{2 c d e}+c^3 d^3 e^7 x^2 \left (3 c d^2-5 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c^3 d^3 e^7 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^5 d^5 e^6}\)

Input:

Int[(x^3*(d + e*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
 

Output:

(2*a^3*e^3*(c*d^2 - a*e^2)*(d + e*x))/(c^5*d^5*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2]) + ((c^3*d^3*e^7*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d 
*e*x^2])/4 + (c^3*d^3*e^7*(3*c*d^2 - 5*a*e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a* 
e^2)*x + c*d*e*x^2] + ((c^3*d^3*e^7*(c^2*d^4 - 34*a*c*d^2*e^2 + 41*a^2*e^4 
)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 - (c^2*d^2*e^7*(((3*c^3 
*d^6 + 13*a*c^2*d^4*e^2 - 187*a^2*c*d^2*e^4 + 187*a^3*e^6)*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])/e - (3*(c*d^2 - a*e^2)*(c^3*d^6 + 5*a*c^2*d^ 
4*e^2 + 35*a^2*c*d^2*e^4 - 105*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x 
)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])] 
)/(2*Sqrt[c]*Sqrt[d]*e^(3/2))))/4)/(2*c*d*e))/(8*c*d*e))/(c^5*d^5*e^6)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4840\) vs. \(2(399)=798\).

Time = 3.09 (sec) , antiderivative size = 4841, normalized size of antiderivative = 11.18

method	result	size
default	\(\text {Expression too large to display}\)	\(4841\)

Input:

int(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN 
VERBOSE)
 

Output:

d^3*(x^2/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/d 
/e/c*(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d 
/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c 
*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d 
^2)*x+c*d*x^2*e)^(1/2))+1/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^( 
1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))-2*a/c*(-1/d/e 
/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+ 
a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^ 
2*e)^(1/2)))+e^3*(1/4*x^5/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-9/ 
8*(a*e^2+c*d^2)/d/e/c*(1/3*x^4/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/ 
2)-7/6*(a*e^2+c*d^2)/d/e/c*(1/2*x^3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e 
)^(1/2)-5/4*(a*e^2+c*d^2)/d/e/c*(x^2/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2 
+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+1/d/e/c*ln((1/2*a*e^2+ 
1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/ 
(d*e*c)^(1/2))-2*a/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a* 
e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/( 
a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)))-3/2*a/c*(-x/d/e/c/(a*d*e+(a*e^...
 

Fricas [A] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 952, normalized size of antiderivative = 2.20 \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="fricas")
 

Output:

[1/256*(3*(a*c^4*d^8*e + 4*a^2*c^3*d^6*e^3 + 30*a^3*c^2*d^4*e^5 - 140*a^4* 
c*d^2*e^7 + 105*a^5*e^9 + (c^5*d^9 + 4*a*c^4*d^7*e^2 + 30*a^2*c^3*d^5*e^4 
- 140*a^3*c^2*d^3*e^6 + 105*a^4*c*d*e^8)*x)*sqrt(c*d*e)*log(8*c^2*d^2*e^2* 
x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^ 
2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c 
*d*e^3)*x) + 4*(16*c^5*d^5*e^4*x^4 - 3*a*c^4*d^7*e^2 - 13*a^2*c^3*d^5*e^4 
+ 315*a^3*c^2*d^3*e^6 - 315*a^4*c*d*e^8 + 24*(c^5*d^6*e^3 - a*c^4*d^4*e^5) 
*x^3 + 2*(c^5*d^7*e^2 - 22*a*c^4*d^5*e^4 + 21*a^2*c^3*d^3*e^6)*x^2 - (3*c^ 
5*d^8*e + 11*a*c^4*d^6*e^3 - 119*a^2*c^3*d^4*e^5 + 105*a^3*c^2*d^2*e^7)*x) 
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^7*d^7*e^3*x + a*c^6*d^6*e 
^4), -1/128*(3*(a*c^4*d^8*e + 4*a^2*c^3*d^6*e^3 + 30*a^3*c^2*d^4*e^5 - 140 
*a^4*c*d^2*e^7 + 105*a^5*e^9 + (c^5*d^9 + 4*a*c^4*d^7*e^2 + 30*a^2*c^3*d^5 
*e^4 - 140*a^3*c^2*d^3*e^6 + 105*a^4*c*d*e^8)*x)*sqrt(-c*d*e)*arctan(1/2*s 
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqr 
t(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2 
*(16*c^5*d^5*e^4*x^4 - 3*a*c^4*d^7*e^2 - 13*a^2*c^3*d^5*e^4 + 315*a^3*c^2* 
d^3*e^6 - 315*a^4*c*d*e^8 + 24*(c^5*d^6*e^3 - a*c^4*d^4*e^5)*x^3 + 2*(c^5* 
d^7*e^2 - 22*a*c^4*d^5*e^4 + 21*a^2*c^3*d^3*e^6)*x^2 - (3*c^5*d^8*e + 11*a 
*c^4*d^6*e^3 - 119*a^2*c^3*d^4*e^5 + 105*a^3*c^2*d^2*e^7)*x)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^7*d^7*e^3*x + a*c^6*d^6*e^4)]
 

Sympy [F]

\[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (d + e x\right )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**3*(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral(x**3*(d + e*x)**3/((d + e*x)*(a*e + c*d*x))**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[6,6,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,0] 
%%%}+%%%{
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\left (d+e\,x\right )}^3}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int((x^3*(d + e*x)^3)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)
 

Output:

int((x^3*(d + e*x)^3)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {x^3 (d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (e x +d \right )^{3}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}d x \] Input:

int(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

int(x^3*(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)